Companies in the area of fractal antennas have been created in the last twenty years hiring many young innovative engineers to develop products for the general economy and the defense industry. Fractal designs in other areas such as archaeology, energy, trade and biology have been instrumental in re-starting the economy and in creating much needed jobs.

Cell phones have impacted society like no other invention. They have been one of the essential ingredients in social relations among family members, friends, lovers and business associates. It is getting more and more difficult to be in a normal relationship in our society without a cell phone.

**Engineering Design Process (EDP)**

**Elements** of the EDP:

§ Identify the problem: To create a fractal design that fits the given criteria.

§ Criteria:

- Must have at least 3 stages
- Must fit within the given plastic sheet
- Must complete an electrical circuit with pos. and neg. connectors (no shorts).
- Team must have an accurate algebraic or recursive formula for the length of the entire fractal and its line segments.

§ Gather information: Use the textbook and internet to view design possibilities and find alternative designs that fit the criteria.

§ Decide which fractal design is the best.

§ Draw the design and plan building the fractal.

§ Write a report:

- Did the design fit the criteria?
- What modifications need to be made on the second try?

§ Improve on the fractal design and re-do this time to 5 stages.

- New fractal design may have to be made to fit the plastic in 5 stages instead of 3.

§ Write a second report:

- Did the design fit the criteria this time?
- Was the second design better than the first?
- What made the second design better or worse?
- If there was a third try what modifications would your team make?

**Iteration** Criteria and amount of time set aside for iteration in the EDP:

- The fractal design must fit the definition of a fractal.
- The design must encompass the most length within a given area.
- The fractal design must be able to complete a circuit.
- A model of the design must be made with at least 3 stages (5 stages on 2
^{nd} try).
- Must be finished within 102 minute double bell (51 minutes on 2
^{nd} try).
- Mathematical equations must be written that accurately generate with a given stage number the………
- Number of segments
- Length of each segment
- Total length of the fractal

##### Unit Academic Standard

9-12.A.CED.1 Create equations that describe numbers or relationship. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*

G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle: bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisectors of a line segments; and constructing a line parallel to a given line through a point not on the line.

G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

G.STR.1 Verify experimentally the properties of dilations given by a center and a scale factor:

a. Dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center changed.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

##### Unit Activities

Activity 1: Fractals, Chaos and the Challenge

Activity 2: Pre-Test

Activity 3: Fractals, Geometry and Self-Similarity

Activity 4: Fractals, Fractions and Algebra

Activity 5: Antenna Science

Activity 6: Building a Fractal Antenna

Activity 7: Post-Test

##### Where the CBL and EDP appear in the Unit

- Challenge Based Learning (CBL) is found in Lesson #1, Activities #3 and #4 and Lesson #2, Activities #6,#7 and #8.
- Engineering Design Process (EDP) is found in Lesson #2, Activities #7 and #8.

##### Misconceptions

- Self-similarity is the same as similarity. Students are taught that objects which are similar could have different proportions. Self-similarity because of the iteration cycle always have the same proportion from the previous stage.
- That mathematics has nothing to do with cell phones.
- That mathematics and science are two distinct different areas of knowledge.

##### Pre-Unit Assessment Instrument

**How to Make This a Hierarchical Unit**

Common Core standards that can be used in a unit for fractals in 8^{th} grade:

- Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
- Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two dimensional figures, describe a sequence that exhibits the similarity.

Activities that could be taught in the 8^{th} grade:

- Drawing fractals
- Proportion problems
- Activities on similar objects
- EDP on building cell phone antennas

**Reflection**

To pass a quarter I require my students to pass a minimum number of tests or projects by 80%. I do this to motivate my students to acquire depth in at least a few areas of math, instead of just getting by with a 60% average, which is not uncommon among adolescents.

It is not unusual in my classroom culture for students to come in after school to re-take a test to acquire the minimum requirements for passage, even though they have the minimum 60% for passage.

It had occurred to me that significantly fewer students came in that quarter to retake the fractals test than other tests. A greater population had reached the 80% threshold on that test, than other tests that quarter.

Subjectively that could be attributed to at least two attributes of CBL (Challenge Base Learning). One is the repetitive nature of the EDP (Engineering Design Process). It was the first time I have lead the students to redesign a project and do it again. In the video testimonies I noticed that the kids talked about, unsolicited from me, that at first they didn't understand, but then when they did it again there was an epiphany.

The other possibility was that they bought into the project. I noticed that they liked the freedom of designing within parameters. They also liked learning and improving upon their teacher's exemplar at the beginning (I reported at the beginning of the project that I made the mistake of building my fractal antenna from the outside in. Not one student team repeated my mistake). They also liked improving on their own designs the second time around.

They had the same experience with the math. I believe that it was important that the groups who chose more complex designs and therefore more difficult math were complimented for taking the risk. In spite of the fact that they had more failures than those who chose easier designs.